
Domain and Range
Hi,
I'm just wondering if I'm correct and also what notation I should use here.
The question is asking for the domain and range of $\displaystyle f(x,y)=e^{yx^2}$
I think the domain is $\displaystyle Domain \subset R^2$ and $\displaystyle Range>0$
Thanks for your help.

Re: Domain and Range
yes, the domain is a subset of $\displaystyle \mathbb{R}^2$. but in this case, if f(x,y) makes sense for all real numbers x and y,
then the domain equals $\displaystyle \mathbb{R}^2$.
now $\displaystyle e^x$ is always positive, so we know the range is a subset of the positive reals. the question is, can $\displaystyle yx^2$
turn out to be any real number? and it can: let's say we have a certain real number r in mind. then we could pick x = 0, and y = r.
so now we ask, given that $\displaystyle yx^2$ can be any real number, can $\displaystyle e^{yx^2}$ be any positive real number?
suppose s is positive. then ln(s) makes sense, so we can pick (x,y) = (0,ln(s)), and then $\displaystyle e^{\ln(s)0} = e^{\ln(s)} = s$.
so we see any positive real number is in the range of f, and only positive real numbers are in the range of f.
so range(f) = $\displaystyle \{x \in \mathbb{R}: x > 0\}$. this is sometimes also written: $\displaystyle \mathbb{R}^+$ or as the interval $\displaystyle (0,\infty)$

Re: Domain and Range
Not sure what your answer is for the Domain: it's all of $\displaystyle \mathbb{R}^2$.
Your Range is correct.
(Deveno beat me by less than 1 minute.)